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Authors: Sergey Fomel, WEI XIE
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Assignment 7

Due in class on Thursday, October 31

Problem 1. Assuming that the following function is periodically extended, find its representation in a Fourier series

f(x)={π+x  ,πx<0πx  ,0x<πf(x) = \displaystyle \left\{\begin{array}{lcr} \pi+x\;,&\quad&-\pi \le x < 0 \\ \pi-x\;,&\quad& 0 \le x < \pi\end{array}\right.

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Problem 2. The Fourier transform is defined as

F(ω)=f(t)eiωtdt\displaystyle F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt

with the inverse transform given by

f(t)=12πF(ω)eiωtdω\displaystyle f(t)=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega;.

The Ricker wavelet, also known as the Mexican hat wavelet, is a popular representation of seismic signals. The Ricker wavelet r(t)r(t) is defined as the second derivate of the Gaussian

r(t)=d2dt2g(t)r(t)=\displaystyle -\frac{d^2}{d t^2} g(t), where g(t)=ea2t2g(t)=e^{-a^2\,t^2}.

Find the Fourier transform of the Ricker wavelet and determine:

  1. peak frequency: the frequency of the maximum in the Fourier spectrum;
  2. centroid frequency: the "center of mass" for the positive part of the spectrum;
  3. apparent frequency: the frequency corresponding to the period between two minima in the time domain.
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t=var('t') gauss(t)=exp(-t^2) ricker(t)=-gauss.diff(t,2) plot(ricker(t),t,-5,5,color='red',thickness=3)
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